Classical subjective expected utility

Abstract

We consider decision makers who know that payoff-relevant observations are generated by a process that belongs to a given class M, as postulated in Wald [Wald A (1950) Statistical Decision Functions (Wiley, New York)]. We incorporate this Waldean piece of objective information within an otherwise subjective setting à la Savage [Savage LJ (1954) The Foundations of Statistics (Wiley, New York)] and show that this leads to a two-stage subjective expected utility model that accounts for both state and model uncertainty.C onsider a decision maker who is evaluating acts whose outcomes depend on some verifiable states, that is, on observations (workers' outputs, urns' drawings, rates of inflation, and the like). If the decision maker (DM) believes that observations are generated by some probability model, two sources of uncertainty affect his evaluation: model uncertainty and state uncertainty. The former is about the probability model that generates observations, and the latter is about the state that obtains (and that determines acts' outcomes).State uncertainty is payoff relevant and, as such, it is directly relevant for the DM's decisions. Model uncertainty, in contrast, is not payoff relevant and its role is instrumental relative to state uncertainty. Moreover, models cannot always be observed: Whereas in some cases they have a simple physical description (e.g., urns' compositions), often they do not have it (e.g., fair coins). For these reasons, the purely subjective choice frameworks à la Savage (1) focus on the verifiable and payoff-relevant state uncertainty. They posit an observation space S over which subjective probabilities are derived via betting behavior.In contrast, classical statistical decision theory à la Wald (2) assumes that the DM knows that observations are generated by a probability model that belongs to a given subset M, whose elements are regarded as alternative random devices that nature may select to generate observations. [As Wald (ref. 2, p. 1) writes, "A characteristic feature of any statistical decision problem is the assumption that the unknown distribution FðxÞ is merely known to be an element of a given class Ω of distributions functions. The class Ω is to be regarded as a datum of the decision problem."] In other words, Wald's approach posits a model space M in addition to the observation space S. In so doing, Wald adopted a key tenet of classical statistics, that is, to posit a set of possible data-generating processes (e.g., normal distributions with some possible means and variances), whose relative performance is assessed via available evidence [often collected with independent identically distributed (i.i.d.) trials] through maximum-likelihood methods, hypothesis testing, and the like. Although models cannot be observed, in Wald's approach their study is key to better understanding state uncertainty.Is it possible to incorporate this Waldean key piece of objective information within Savage's framework? Our work addresses this question and tries to embed this classical datum wi...

Managing Catastrophic Climate Risks Under Model Uncertainty Aversion

“…In this paper, we take a step further in the direction of understanding the theoretical mechanisms underlying the results obtained in this literature, by applying the most recent robust tools developed in decision theory (Cerreia-Vioglio et al 2013b, Marinacci 2015. We consider an alternative to expected utility models that allows us to incorporate both risk and model uncertainty.…”

“…Cerreia-Vioglio et al (2013b) are the first to provide a decision theoretic derivation of this type of deep uncertainty presented in two layers. In particular, they enrich the standard Savage framework (Savage 1954) in the presence of objective information, and show that preferences satisfying Savage's axioms may be represented, in the context of our abatement model, by…”

Ellsberg Re-Revisited: An Experiment Disentangling Model Uncertainty and Risk Aversion

“…It is then expressed in monetary terms by considering the certainty equivalent for each model P θ . Since P θ is itself uncertain, the different certainty equivalents are then evaluated by considering the expected utility using function v. In the case where both attitudes towards the different types of uncertainty are identical (whenever v is equal, up to an affine transformation, to u), we recover the classical subjective expected utility model of Cerreia-Vioglio et al (2013b). This model therefore encompasses both the Savagian subjective expected utility 15 and the classical von Neumann-Morgernstern representations.…”

Finding Common Ground When Experts Disagree: Robust Portfolio Decision Analysis

“…We note that Subjective Expected Utility using averaged probabilities (SEUa from now on, (Cerreia-Vioglio et al 2013)) can be regarded as a special case of KMM.…”